Download Countability Axioms for Topological Spaces

Introduction
Countable Space
Results on Countable Spaces
Appendix
Lecture : Countability Axioms
for
Topological Spaces
Dr. Sanjay Mishra
Department of Mathematics
Lovely Professional University
Punjab, India
October 28, 2014
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Outline
1
Introduction
2
Countable Space
First Countable Space
Second Countable Space
Lindelof Space
3
Results on Countable Spaces
4
Appendix
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Introduction I
The concept which we are going to discuss are unlike compactness and
connectedness that do not arise naturally from the study of calculus nd
analysis. This concept is for deeper study of topology itself.
Such problems as imbedding a given space in a metric space or in compact Housdorff space are basically problems of topology rather than analysis. These particular problems have solutions that involve the countability axioms.
Our basic goal is to prove the “Urysohn metrization theorem”. That is if
a topological space X satisfy a certain countability axiom (second) and
a certain separation axiom (regular), then the X can be imbedded in
a metric space and is thus metrizable. So, countability axiom will play
very important role in proving of this theorem.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
First Countable Space I
Definition (First Countable Space)
A topological space X is called first countable space is it satisfies the
following axiom called the first axiom of countability. “ For each point
p ∈ X there exists a countable base Bp of open sets containing p such
that every open set G containing p also contains a member of Bp .
Or in other words, a topological space X is first countable space if and
only if there exists a countable local base at every point p ∈ X.
Remark
The first countable axiom is a local property of a topological space X,
i.e. if depends only upon the properties of arbitrary neighborhoods of
the point p ∈ X.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
First Countable Space II
Example
Let X be metric space and let p ∈ X. Recall that the countable class of
open spheres {S(p, 1), S(p, 12 ), S(p, 13 ), . . .} with center p is a local base
at p. Hence every metric space satisfies the first axiom of countability.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
First Countable Space III
Example
Let X be any discrete space. Now the singleton set {p} is open and is
contained in every open set G containing p ∈ X. hence every discrete
space satisfies first axiom of countability.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
First Countable Space IV
Example
Let X = N and J = {X, φ, {1}, {1, 2}, . . . , {1, 2, . . . , n}, . . .} then
obviously (X, J ) is a first countable topological space.
Note that this is not an interesting example of a first countable topological space. Once the topology J is a countable collection then (X, J ) is
a first countable space.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
First Countable Space V
Example
Let X = R and Jl be the lower limit topology on R generated by
{[a, b) : a, b ∈ R, a < b}. For each x ∈ X, Bx = {[x, x + n1 ) : n ∈ N} is a
countable local base at x. Hence (R, Jl ) = Rl is a first countable
topological space.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
Second Countable Space I
Definition (Second Countable Space)
A topological space (X, τ ) is called a second countable space if it
satisfies the following axiom, called second axiom of countability.
“There exists a countable base B for the topology τ ”.
Remark
Second countability is a global rather than a local property of a
topological space.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
Second Countable Space II
Example
The class B of open intervals (a, b) with rational endpoints, i.e a, b ∈ Q,
is countable and is a base for the usual topology on the real line R.
Thus R is a second countable space.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
Second Countable Space III
Example
Consider the discrete topology D on R. Recall that a class B is base for
a discrete topology if an only if if contains all singleton sets. But R,
and hence the class of singleton subsets {p} of R, are non-countable.
Accordingly, (R, D) does not satisfy the second axiom of countability.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
Second Countable Space IV
Comparison between first and second countable spaces
First countable space
For each point p ∈ X there exists a
countable base Bp of open sets
containing p such that every open
set G containing p also contains a
member of Bp .
Second countable space
There exists a countable
base B for the topology τ .
It is global property of
space.
It is local property of space.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
Lindelof Space I
A Lindelof space is a topological space in which every open cover has a
countable subcover. The Lindelof property is a weakening of the more
commonly used notion of compactness, which requires the existence of
a finite subcover.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
First Countable Space
Second Countable Space
Lindelof Space
Lindelof Space II
Definition
A topological space (X, τ ) is said to be aSLindelof space if for any
collection A of open sets such that X = A∈A A there
S exists a
countable subcollection say B ⊂ A such that X = B∈B B.
or we can say that (X, τ ) is called Lindelof space f every cover of X is
reducible to a countable cover.
From this definition it show that
“ Every compact space is Lindelof space but its converse is not true”.
For example a discrete topological space (X, τ ) is Lindelof space where
X is any infinite countable set but it is not compact.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces I
Now here we will discuss some important results on countable spaces as
like,
Relation between first and second countable spaces.
Hereditary property of countable space.
Subspace of countable space.
Countable product of topological spaces.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces II
Theorem
A second countable space is always first countable space.
Proof:
Here we want to show that
Second countable space ⇒ First countable space.
As given that space (X, τ ) is second countable space, then
⇒∃ a countable base B for τ on X
⇒B ∼ N
⇒B can be expressed as B = {Bn ∈ B : x ∈ Bn }
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces III
Let x ∈ X be arbitrary and a new collection of members of B is
Lx = {Bn ∈ B : x ∈ Bn }
Now for Lx we can say that
1
Since subset of countable set B is countable so the collection Lx
will be countable.
2
Because Lx ⊂ B and members of B are τ -open sets, so members of
Lx will be also τ -open sets.
3
Any G ∈ Lx ⇒ x ∈ G, according to the construction of Lx .
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces IV
4
Let G ∈ τ be arbitrary such that x ∈ G. Then, by definition of
base
x ∈ G ∈ τ ⇒Br ∈ B such that x ∈ Br ⊂ G
⇒∃ Br ∈ Lx such that Br ⊂ G
For Br ∈ B with x ∈ Br ⇒ Br ∈ Lx .
Finally, we can say that
x ∈ G ∈ τ ⇒ ∃ Br ∈ Lx such that Br ⊂ G
Hence we can say that Lx is a countable local base at x ∈ X. So,
by definition X is first countable space.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces V
Theorem
First countable space does not imply second countable space.
Proof:
Let τ be a discrete topology on infinite set X so that every subset of X
is open in X and hence in particular, each singleton set {x} is open in
X for each x ∈ X. Write B = {{x} : x ∈ X}.
Here we can verify that B is a base for τ on X and B is uncountable.
For X is uncountable. Hence X is not second countable space.
If we take Lx = {x}, then evidently Lx is a countable local base at x ∈ X
as it has only one member.
For any G ∈ τ with x ∈ G, there exists {x} such that x ∈ {x} ⊂ G.
From definition, X is first countable but not second countable space.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces VI
Here we will discuss the hereditary property of countable spaces.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces VII
Theorem
The property of a space being first countable is hereditary.
Proof:
Let (Y, τ1 ) be a subspace of first countable (X, τ ), then we want to prove
that (Y, τ1 ) is also first countable space.
Let y ∈ Y ⊂ X be arbitrary. If X is first countable space show that
there exists a countable local base x ∈ X. Consider countable local base
B = {Bn : n ∈ N} for y ∈ X. Then y ∈ Bn , ∀ n ∈ N.
Write
B1 = {Y ∩ Bn : n ∈ N}
(1)
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces VIII
y ∈ Y, y ∈ Bn ∀ n ∈ N ⇒y ∈ Y ∩ Bn , ∀ n ∈ N
Bn ∈ B ∀ n ∈ N ⇒Bn ∈ τ ⇒ Y ∩ Bn ∈ τ1
Now we claim that B1 is a countable local base at y for τ1 on Y .
1
Since by equation (2) there is bijection map n → Y ∩ Bn exists
between N and B1 , so N ∼ B1 . Hence B1 is countable.
2
For any G ∈ B1 ⇒ y ∈ G.
3
The collection B1 is family of all τ1 -open sets.
Sanjay Mishra
Countability Axioms
(2)
(3)
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces IX
4
Let G ∈ τ1 be arbitrary such that y ∈ G, then there exists H ∈ τ
such that G = H ∩ Y , (by definition of subspace).
As assume that y ∈ Y , y ∈ G and G = H ∩ Y ⇒ y ∈ H.
By definition of local base for X
y ∈ H ∈ τ ⇒∃ Br ∈ B such that y ∈ Br ⊂ H
y ∈ H ∩ Y ∈ T1 ⇒∃ Br ∩ Y ∈ B1 such that Br ∩ Y ⊂ H ∩ Y
y ∈ G ∈ τ1 ⇒Br ∩ Y ∈ B1 such that y ∈ Br ∩ Y ⊂ G
(4)
Finally we can say that B1 is a local base at y ∈ Y for topology τ1
on Y , hence (Y, τ1 ) is first countable space.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces X
Theorem
The property of a space being second countable space is hereditary.
Proof:
Let (Y, τ1 ) be a subspace of topological space (X, τ ) which is second
countable space. We want to proof that Y is also second countable
space.
By given, there exists a countable base B for τ on X.
B is countable ⇒ B ∼ N ⇒ B is expressible as B = {Bn : n ∈ N}
Write
B1 = {Y ∩ Bn : n ∈ N}
(5)
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces XI
1
Since by equation (5) there is bijection map n → Y ∩ Bn exists
between N and B1 , so N ∼ B1 . Hence B1 is countable.
2
The collection B1 is family of all τ1 -open sets.
For Bn ∈ B ⇒Bn ∈ τ
∵B⊂τ
⇒Y ∩ Bn ∈ τ1 ∵ Y is subspace of X.
3
Any y ∈ G ∈ τ1 ⇒ ∃ Br ∩ Y ∈ B1 such that y ∈ Y ∩ Br ⊂ G.
For proving this let G ∈ τ1 such that y ∈ G, then there exists
H ∈ τ such that G = H ∩ Y .
y ∈ G ⇒ y ∈ H ∩ Y ⇒ y ∈ H, y ∈ Y
By definition of base, any y ∈ H ∈ τ ⇒ ∃ Br ∈ B such that
y ∈ Br ⊂ H
Sanjay Mishra
Countability Axioms
(6)
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces XII
From which
any y ∈ H ∩ Y ∈ τ1 ⇒ ∃ Y ∩ Br ∈ B1 such that y ∈ Y ∩ Br ⊂ G.
i.e. any y ∈ G ∈ τ1 ⇒ ∃ Y ∩ Br ∈ B1 such that y ∈ Y ∩ Br ⊂ G.
From the definition and above results we can say that B1 is a countable
base for τ1 on Y . Consequently (Y, τ1 ) is second countable space.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces XIII
Theorem
A second countable space is Lindelof space.
Or Let (X, τ ) is second countable space. Let O an open cover of a set
A ⊂ X. Then O is reducible to a countable cover.
Proof:
As given (X, τ ) is a second countable space and O be an open cover
A ⊂ X, so that
A ⊂ ∪{O : O ∈ O}
(7)
Here we want to show that O is reducible to a countable cover.
As X is countable space so, there exists a countable base B for τ on X.
Since B is countable and hence its members may be enumerated as
B1 , B2 , . . ..
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Results on Countable Spaces XIV
Let x ∈ A be arbitrary, then equation (7) ⇒ x ∈ Ox for at least one
Ox ∈ O. Since Ox ⊂ X is open, then Ox ∈ τ .
By definition of base
x ∈ Ox ∈ τ ⇒ ∃Bx ∈ B such that x ∈ Bx ⊂ Ox
We can write
A ⊂ ∪{Bx : x ∈ A}
(8)
(9)
∴ ∪{Bx : x ∈ A} ⊂ B and hence it can be expressed as
A ⊂ ∪{Bn : n ∈ N}
(10)
Choose On ∈ O such that Bn ⊂ On ∀n ∈ N
⇒
∪{Bn : n∈N } ⊂ ∪{On : n ∈ N}
In this event (10) reduces to A ⊂ ∪{On : n ∈ N}
This show that the open cover O of A reducible to a countable cover
{On : n ∈ N}.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Definition (Finite and Infinite Set)
A set X is finite if X is empty or if there is a bijection
f : {1, 2, . . . , n} → X for some n ∈ Z+ .
A set that is not finite is said to be infinite.
A special type of infinite set is the countably infinite set, a prototype of
which is the set of positive integers Z+ .
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Definition (Countable and Uncountable Set)
A set X is countably infinite if there is a bijection f : Z+ → X. A set
that is either finite or countably infinite is said to be countable.
A set that is not countable is said to be uncountable.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Definition (Local Base)
Let (X, τ ) be a topological space. A family Bx of open subsets of X is
said to be a local base at x ∈ X for τ on X if
1
Any B ∈ Bx ⇒ x ∈ B.
2
Any G ∈ τ with y ∈ G ⇒ ∃ B ∈ Bx such that y ∈ B ⊂ G.
Example
Let x ∈ R be arbitrary. And
An = (x − n1 , x + n1 ) ∀ n ∈ N
Take Bx = {An : n ∈ N}. Evidentaly Bx is a local base at x ∈ R for
usual topology on R.
Sanjay Mishra
Countability Axioms
Introduction
Countable Space
Results on Countable Spaces
Appendix
Theorem
1
A subset of a finite set is a finite set.
2
A finite union of finite sets is a finite set.
3
A product of finite sets is a finite set.
4
A subset of a countable set is a countable set.
5
A countable union of countable sets is a countable set.
6
A product of countable sets is a countable set.
Sanjay Mishra
Countability Axioms